Question: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $k \neq 0$. $p = \dfrac{10(4k - 9)}{2} \div \dfrac{20k - 45}{k} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $p = \dfrac{10(4k - 9)}{2} \times \dfrac{k}{20k - 45} $ When multiplying fractions, we multiply the numerators and the denominators. $p = \dfrac{ 10(4k - 9) \times k } { 2 \times (20k - 45) } $ $ p = \dfrac {k \times 10(4k - 9)} {2 \times 5(4k - 9)} $ $ p = \dfrac{10k(4k - 9)}{10(4k - 9)} $ We can cancel the $4k - 9$ so long as $4k - 9 \neq 0$ Therefore $k \neq \dfrac{9}{4}$ $p = \dfrac{10k \cancel{(4k - 9})}{10 \cancel{(4k - 9)}} = \dfrac{10k}{10} = k $